Begin["System`"]

Sin[Pi] := 0
Sin[Pi/2] := 1
Sin[Pi/3] := Sqrt[3]/2
Sin[2 Pi/3] := Sqrt[3]/2
Sin[Pi/4] := Sqrt[2]/2
Sin[3 Pi/4] := Sqrt[2]/2
Sin[Pi/6] := 1/2
Sin[5 Pi/6] := 1/2
Sin[0] := 0
Sin[k_Integer] := -Sin[-k] /; k < 0
Sin[k_Rational] := -Sin[-k] /; k < 0
Sin[k_Integer Pi] := 0
Sin[k_Rational Pi] := -Sin[-k Pi] /; k < 0
(*
Sin[k_Rational Pi] := Module[{n = IntegerPart[k], x = Pretty[k], q, p},
q = x[[1]]; p = x[[2]];
If[n != 0, (-1)^n Sin[((q%p)/p) Pi],'Sin[k Pi]]] /; k > 0
*)
Sin[-x_] := -Sin[x]
Sin[Pi+x_] := -Sin[x]
Sin[k_Integer Pi+x_] := (-1)^k Sin[x]
Sin[Pi/2+x_] := Cos[x]

Cos[Pi] := -1
Cos[Pi/2] := 0
Cos[Pi/3] := 1/2
Cos[2 Pi/3] := -1/2
Cos[Pi/4] := Sqrt[2]/2
Cos[3 Pi/4] := -Sqrt[2]/2
Cos[Pi/6] := Sqrt[3]/2
Cos[5 Pi/6] := -1/2
Cos[0] := 1
Cos[k_Integer] := Cos[-k] /; k < 0
Cos[k_Rational] := Cos[-k] /; k < 0
Cos[k_Integer Pi] := (-1)^k
(*
Cos[k_Rational Pi] := Module[{n = IntegerPart[k], x = Pretty[k], q, p},
q = x[[1]]; p = x[[2]];
If[n != 0, (-1)^n Cos[((q%p)/p) Pi],'Cos[k Pi]]] /; k > 0
*)
Cos[-x_] := Cos[x]
Cos[Pi+x_] := -Cos[x]
Cos[k_Integer Pi+x_] := (-1)^k Cos[x]
Cos[Pi/2+x_] := -Sin[x]

Log[1] := 0
Exp[0] := 1
Exp[x_] := E^x /; Head[x] =!= Real
Log[1] := 0
Log[E] := 1
Log[E^n_] := n

End[]